What gets to be called a "proper class?"

Question

ZFC has no formal notion of "proper class," but informally, everyone uses the term anyway. $V$, $Ord$, etc are said to be proper classes. Similarly, although in ZFC, one can only take the "union" of some collection of sets if the entire collection is also a set, we have no problem saying $V$ is the "union" of the various $V_\alpha$. All of this would seem to be fairly self-evident and harmless.

But there is a peculiar subtlety about which "non-set-collections" of the universe get this label of "proper class," and I am curious to get some clarification. Here is what I mean, ordered from least to most controversial:

• We have the various ordinals. Each one is a set, but their union, $Ord$, isn't a set. Is this union a proper class? (Yes.)
• We have the various sets $V_a$ from the Von Neumann hierarchy. Each of these is a set, but their union, $V$, is not a set. Is this union a proper class? (Yes.)
• Suppose we have ZF with the negation of the axiom of infinity. We have some infinite collection $S$ of natural numbers. Each finite subset of $S$ is a set, but their union, $S$, is not a set. Is this union a proper class? (I think so, relative to this theory.)
• Suppose we have some model of ZFC which is missing some subset of the naturals called $S$. Each finite subset of $S$ is a set, but their union, $S$, is not a set. Is this union a proper class, even though it is now a subclass of a set? (Uh, yes? No?)
• Suppose we have some countable model of ZFC. The model thinks that "$\omega_1$" is uncountable and regular, but there exists some countable set $S$ of ordinals that is cofinal in "$\omega_1$" which the model doesn't know about. Each finite subset of $S$ is a set, but their union, $S$, is not a set. Is this union a proper class, even though it is now a subclass of a set? (Maybe?)
• We are in positive set theory, which has a set of all sets. There are many sets in here which are well-founded, but the union on all of these is not a set. Is this union a proper class, even though it is now a subclass of a set? (I'm sure I've heard the term used this way.)

We can go a lot further, but hopefully the point is clear. There can be collections of elements which fail to be a set for reasons other than being "too large." This happens all the time, particularly in ZF without choice, where there seems to be no limit to how bizarre some of these models can get. Do we think of these things as "proper classes?" Maybe "semisets?"

Answer 1

The term "class" is not a technical term with a universally definite meaning, but there are various established meanings in various contexts.

In ZFC the established usage as Wojowu mentions in the comments is that a class is any definable collection, allowing parameters. But in truth, one commonly also hears the term "definable class" to refer to this situation, especially from those researchers who sometimes routinely also work in the second-order contexts of GBC or KM.

However, it happens at times that one has a model $M$ of ZFC and subcollection of the sort you mention, which is not definable, but one wants to add it as an allowed class. An amenable class is one that can be added to the model while preserving certain parts of the theory. A weak form of amenability for a class $A$ should mean that $A\cap a$ is a set in $M$ for every $a\in M$. A stronger form would meant that $\langle M,\in,A\rangle$ satisfies the replacement axiom in the language allowing $A$. This is sometimes also called ZFC-amenable.

For example, class forcing often works this way over ZFC, since one often wants to augment the generic filter (which is not definable) to the model, and indeed, for tame class forcing the generic filter is amenable. This is how one can use class forcing to add an amenable class choice function or an amenable global well order to a model of ZFC.

In second-order set theory $\text{ZFC}_2$, a class means any subcollection of the universe whatsoever.

In Gödel-Bernays set theory GBC, one has specified explicitly the family of classes (for the Henkin semantics), and to have a class means to have a member of that family. It often arises that one will expand the family of classes, such as by class forcing, and so one will specify whether one refers to classes in the original model or the extension.

In finite set theory or arithmetic, the usage is similar to the above. In PA, one would often understand a class to mean a definable class, allowing parameters, but more generally, one considers the arithmetic version of amenability (called an "inductive" class), which means the model satisfies the induction scheme in the language with a predicate for the new class.

In second-order arithmetic, used pervasively in reverse mathematics, one has an explicit family of allowed classes (but they are just called "the sets" as opposed to the numbers), like the GBC case but for arithmetic. In $\text{RCA}_0$, for example, one has for sure only the computable sets as classes, but in $\text{ACA}_0$ one will have all arithmetically definable classes. In full second-order arithmetic, we allow arbitrary subsets of the model.

We often don't use the word "class" to describe a subcollection of the universe that would destroy important parts of the theory in which we are interested. For example, we might have a cut in a model of PA, but nontrivial instances of this are never inductive, and we don't usually call them classes. In this sense, we don't tend to use the word "class" in the weirder examples you provide.